Problem: All of the 5th grade teachers and students from Almond went on a field trip to an archaeology museum. Tickets were $$8.00$ each for teachers and $$4.50$ each for students, and the group paid $$77.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$24.00$ each for teachers and $$12.50$ each for students, and the group paid $$221.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8x+4.5y = 77}$ ${24x+12.5y = 221}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-24x-13.5y = -231}$ ${24x+12.5y = 221}$ Add the top and bottom equations together. $ -y = -10 $ $ y = \dfrac{-10}{-1}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {8x+4.5y = 77}$ to find $x$ ${8x + 4.5}{(10)}{= 77}$ $8x+45 = 77$ $8x = 32$ $x = \dfrac{32}{8}$ ${x = 4}$ You can also plug ${y = 10}$ into $ {24x+12.5y = 221}$ and get the same answer for $x$ ${24x + 12.5}{(10)}{= 221}$ ${x = 4}$ There were $4$ teachers and $10$ students on the field trips.